Pricing of Basket Options
Chapter One
Preamble of the Study
One of the more extensively sold exotic options is the basket option, an option whose payoff depends on the value of a portfolio or basket of assets. At maturity it pays off the greater of zero and the difference between the average of the prices of the n different assets in the basket and the exercise price.
The typical underlying of a basket option is a basket consisting of several stocks, that represents a certain economy sector, industry or region.
The main advantage of a basket option is that it is cheaper to use a basket option for portfolio insurance than to use the corresponding portfolio of plain vanilla options. Indeed, a basket option takes the imperfect correlation between the assets in the basket into account and moreover the transaction costs are minimized because an investor has to buy just one option instead of several ones.
CHAPTER TWO
LITERATURE REVIEW
A lot of research has been done on the pricing of basket options, involving the use of different methods. Cox(1979) was the first to propose the Tree based approach, adopted in Wan(2002). Basket option pricing is done by ap- proximating the underlying basket distribution. It employs the conditional expectation method first suggested by Curran(1994), Rogers and Shi(1995) and Nielsen and Sandmann (2003) for Asian options. Beisser(1999) esti- mated the price of the basket call from the weighted sum of (artificial) European call prices.
Gentle (1993) approximates the arithmetic average of the basket pay-off by a geometric average. Levy (1992) approximate the distribution of the basket by a log-normal distribution. Ju (2002) considers a Taylor expansion of the ratio of the characteristic function of the arithmetic average to that of the approximating log-normal random variable around zero volatility and provides a closed-form solution.[6]
The fact that the distribution of correlated log-normally distributed ran- dom variables converges to the reciprocal gamma distribution as the num- ber of the underlying asset approaches infinity makes Milevsky and Pos- ner(1998a) use the reciprocal gamma distribution as an approximation for the distribution of the basket,and provide an analytical solution.[4] The first two moments of both distributions where matched to obtain a closed-form solution. Later (1998b) they use distributions from the Johnson (1994) fam- ily as state-price densities to match higher moments of distribution of the arithmetic mean. Cox and Ross (1976) noted that if a risk-less hedge can be formed, the option value is the risk-neutral and discounted expectation of its pay-off.
Boyle(1977)[31] was the first to propose the Monte Carlo methods to option pricing, as an alternative to the previous methods.This draws the attention of researchers on the use of Monte Carlo method, and bed to the introduction of some variance reduction techniques. The price is estimated by simulating many independent paths of the underlying assets and taking the discounted mean of the generated play-offs. Boyle (1977), Kemna and Vorst (1990), Clewlow and Carverhill (1993), wrote some papers on variance reduction of the estimates of option prices. Ripley( 1987) and Hammersley and Handscomb (1967) wrote on the use of control variates to reduce the variance of Monte Carlo estimates. P. Pellizzari (1997) presents two kinds of control variates to reduce variance of estimates, based on unconditional and conditional expectations of assets.
In a copula framework, an upper bound on a basket option is obtained by Rapuch and Roncalli (2001) and Cherubini and Luciano (2002). It is shown that this bound is equal to the so-called Frechet bound and corre- sponds to a particular case where the underlying assets are co-monotonic. Chen, Deelstra, Dhaene and Vanmaele (2006) use the related idea based on the theory of stochastic orders and on the theory of comonotonic risks, to derive the largest possible price that occurs when the components assets are comonotonic. Lars Oswald Dahl and Fred Espen Benth wrote on valuation of Asian Basket Options with Quasi-Monte Carlo Techniques and Singu- lar Value Decomposition. Jinke Zhou1 and Xiaolu Wang (2008) provided a closed-form approximation formulae for pricing basket options. By ap- proximating the distribution of the sum of correlated lognormals with some log-extended-skew-normal distribution.
CHAPTER THREE
FINANCIAL DERIVATIVES
A financial derivative may be defined as a financial instrument whose value depends on (or derives from) the values of other, more basic underlying variables (assets, securities or indices) or is a financial contract whose value at an expiration date T, which is written into the contract, is determine by the price stochastic process of some financial asset called the underlying financial asset up to time T. Financial derivatives are traded in derivative ex- changes (markets where individuals trade standardized contracts that have been defined by the exchange) and over the counter markets (a telephone and computer linked network of dealers,who do not physically meet). To every contract there are two positions: long position (the buyer of the con- tract)and short position (the seller/writer of the contract).[35]
CHAPTER FOUR
PRICING OF BASKET OPTION
In this chapter we discuss the pricing of basket options, factors that affect the pricing and various pricing techniques.
A lognormal distribution is the distribution of choice when pricing assets. There are three important observations that we have to recognize in order to understand this: the price of a stock at time t1 is dependent on two variables: the stock price at t0 and rate of return (r) for the interval [t0, t1]: What this means is that for one period of time [t0, t1], we can get the new stock price by using the simple transformation:
past stock price × rate of return = current stock price
The rate of return (r) follows a normal distribution: this is an important assumption. In order to completely understand this we should first think about the properties of a normal distribution, particularly: the shape of the distribution. Just by looking at a normal distribution graph, we can easily tell that the farther you move away from the mean the likelihood of getting a sample becomes less hence the likelihood of getting samples close to the mean is much higher. In other words: if we have a mean rate-of-return equal to r0 for time t1, we will see the new rate-of-return r1 at time t1 very ”close to” r0. This property of being ”close to” gives us the idea of the rate-of- return conforming to a normal distribution. For the time interval [t0, t1] we continuously compound the return: a very important concept in finance is compounding the return over an interval of time.
A continuously compounded rate of return is expressed using the math- ematical concept of exponent ex. So, if we have a rate of return = r, then the continuously compounded rate of return is: er.
Chapter Five
Conclusion
As the time increases the option value increases.
We have successfully described some of the methods used in pricing basket options. We considered the European type, which is easier to price than an American type, which allows the buyer to exercise the option on or before expiration. Further research can focus on basket options of American type.
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- Niklas Ekvall(1995):A lattice approach for pricing of multivariate con- tingent claimsEuropean Journal Of Operational
- Nengjiu Ju(2000):Pricing Asian and Basket Options Via Taylor Expan- sion.The Robert Smith School of Business University of Maryland.
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